Critical magnetic flux for Weyl points in the three-dimensional Hofstadter model

We investigate the band structure of the three-dimensional Hofstadter model on cubic lattices, with an isotropic magnetic field oriented along the diagonal of the cube with flux Φ=2⁢𝜋𝑚/𝑛, where 𝑚,𝑛 are coprime integers. Using reduced exact diagonalization in momentum space, we show that, at fixed 𝑚, there exists an integer 𝑛⁡(𝑚) associated with a specific value of the magnetic flux, that we denote by Φ𝑐⁡(𝑚)≡2⁢𝜋𝑚/𝑛⁡(𝑚), separating two different regimes. The first one, for fluxes Φ<Φ𝑐⁡(𝑚), is characterized by complete band overlaps, while the second one, for Φ>Φ𝑐⁡(𝑚), features isolated band-touching points in the density of states and Weyl points between the 𝑚⁢th and the (𝑚+1)-th bands. In the Hasegawa gauge, the minimum of the (𝑚+1)-th band abruptly moves at the critical flux Φ𝑐⁡(𝑚) from 𝑘𝑧=0 to 𝑘𝑧=𝜋. We then argue that the limit for large 𝑚 of Φ𝑐⁡(𝑚) exists and it is finite: lim𝑚→∞⁡Φ𝑐⁡(𝑚)≡Φ𝑐. Our estimate is Φ𝑐/2⁢𝜋=0.1296⁢(1). Based on the values of 𝑛⁡(𝑚) determined for integers 𝑚≤60, we propose a mathematical conjecture for the form of Φ𝑐⁡(𝑚) to be used in the large-𝑚 limit. The asymptotic critical flux obtained using this conjecture is Φ(conj)𝑐/2⁢𝜋=7/54.